Hölder-type inequalities and their applications to concentration and correlation bounds

Abstract

Let $Y_v, v\in V,$ be $[0,1]$-valued random variables having a dependency graph $G=(V,E)$. We show that \[ \mathbb{E}\left[\prod_{v\in V} Y_{v} \right] \leq \prod_{v\in V} \left\{ \mathbb{E}\left[Y_v^{\frac{χ_b}{b}}\right] \right\}^{\frac{b}{χ_b}}, \] where $χ_b$ is the $b$-fold chromatic number of $G$. This inequality may be seen as a dependency-graph analogue of a generalised Hölder inequality, due to Helmut Finner. Additionally, we provide applications of Hölder-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables.